Ukrainian Mathematical Journal

, Volume 66, Issue 5, pp 645–665 | Cite as

On the Behavior of Algebraic Polynomial in Unbounded Regions with Piecewise Dini-Smooth Boundary

  • F. G. Abdullayev
  • P. Özkartepe
Let G ⊂  be a finite region bounded by a Jordan curve L := ∂G, let \( \Omega :=\mathrm{e}\mathrm{x}\mathrm{t}\overline{G} \) (with respect to \( \overline{\mathbb{C}} \)), let Δ := {w : |w| > 1}, and let w = Φ(z) be the univalent conformal mapping of Ω onto Δ normalized by Φ (∞) = ∞, Φ′(∞) > 0. Also let h(z) be a weight function and let A p (h,G), p > 0 denote a class of functions f analytic in G and satisfying the condition
$$ {\left\Vert f\right\Vert}_{A_p\left(h,G\right)}^p:={\displaystyle \int {\displaystyle \underset{G}{\int }h(z){\left|f(z)\right|}^pd{\sigma}_z<\infty, }} $$
where σ is a two-dimensional Lebesgue measure.
Moreover, let P n (z) be an arbitrary algebraic polynomial of degree at most n ∈ ℕ. The well-known Bernstein–Walsh lemma states that
$$ \begin{array}{cc}\hfill \left|{P}_n(z)\right|\le {\left|\varPhi (z)\right|}^n{\left\Vert {P}_n\right\Vert}_{C\left(\overline{G}\right)},\hfill & \hfill z\in \Omega .\hfill \end{array} $$

In this present work we continue the investigation of estimation (*) in which the norm \( {\left\Vert {P}_n\right\Vert}_{C\left(\overline{G}\right)} \) is replaced by \( {\left\Vert {P}_n\right\Vert}_{A_p\left(h,G\right)},p>0 \), for Jacobi-type weight function in regions with piecewise Dini-smooth boundary.


Orthogonal Polynomial Quasiconformal Mapping Jordan Curve Algebraic Polynomial Unbounded Region 
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Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  • F. G. Abdullayev
    • 1
  • P. Özkartepe
    • 1
  1. 1.Mersin UniversityMersinTurkey

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